It feels like this is in tension with the insensitivity condition, like that this only permits universal generalizations (I guess stuff like anatomy?)
Quite the opposite! In that example, what the insensitivity condition would say is: if I get a big sample of people (roughly 50⁄50 male/female), and quantify the average niceness of men and women in that sample, then I expect to get roughly the same numbers if I drop any one person (either man or woman) from the sample. It’s the statistical average which has to be insensitive; any one “sample” can vary a lot.
That said, it does need to be more like a universal generalization if we impose a stronger invariance condition. The strongest invariance condition would say that we can recover the latent from any one “sample”, which would be the sort of “universal generalization” you’re imagining. Mathematically, the main thing that would give us is much stronger approximations, i.e. smaller ϵ’s.
Quite the opposite! In that example, what the insensitivity condition would say is: if I get a big sample of people (roughly 50⁄50 male/female), and quantify the average niceness of men and women in that sample, then I expect to get roughly the same numbers if I drop any one person (either man or woman) from the sample. It’s the statistical average which has to be insensitive; any one “sample” can vary a lot.
That said, it does need to be more like a universal generalization if we impose a stronger invariance condition. The strongest invariance condition would say that we can recover the latent from any one “sample”, which would be the sort of “universal generalization” you’re imagining. Mathematically, the main thing that would give us is much stronger approximations, i.e. smaller ϵ’s.
Oops, my bad for focusing on the simplified version and then extrapolating incorrectly.